In [1]}, using an analog of Bochner's method or \(``B^2 - 4AC \le 0"\)  method, Wei and Li derive a geometric differential-integral inequality on a manifold (see Theorem REF ) and give the first application of the Hessian comparison theorems to generalized sharp Caffarelli-Kohn-Nirenberg type inequalities on
Riemannian manifolds(see Theorem REF ) . The case \(M=\mathbb {R}^n\)  is due to L. Caffarelli et al. [2]} and is sharp due to Costa [3]}. Furthermore, Wei and Li [1]} also gave the first application of the Hessian comparison theorems to generalized Hardy type inequalities on
Riemannian manifolds. The case \(M=\mathbb {R}^n\)  is sharp (see [5]}, [6]}).
